# Tagged Questions

Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

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### Why $\sqrt{-1 \times {-1}} \neq \sqrt{-1}^2$?

I know there must be something unmathematical in the following but I don't know where it is: \begin{align} \sqrt{-1} &= i \\ \\ \frac1{\sqrt{-1}} &= \frac1i \\ \\ \frac{\sqrt1}{\sqrt{-1}} &...
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### How to prove Euler's formula: $e^{it}=\cos t +i\sin t$?

Could you provide a proof of Euler's formula: $e^{it}=\cos t +i\sin t$ ? thanks.
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### Why $\sqrt{-1 \times -1} \neq \sqrt{-1}^2$? [duplicate]

We know $$i^2=-1$$then why does this happen? $$i^2 = \sqrt{-1}\times\sqrt{-1}$$ $$=\sqrt{-1\times-1}$$ $$=\sqrt{1}$$ $$= 1$$ EDIT: I see this has been dealt with before but at least with ...
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### Do complex numbers really exist?

Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious ...
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### Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$

Using $\text{n}^{\text{th}}$ root of unity $$\large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$ Prove that $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
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### Do odd imaginary numbers exist?

Is the concept of an odd imaginary number defined/well-defined/used in mathematics? I searched around but couldn't find anything. Thanks!
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### What are imaginary numbers?

At school I really struggled to understand the concept of imaginary numbers. My teacher told us that an imaginary number is a number which has something to do with the square root of $-1$. When I ...
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### Prove that $i^i$ is a real number

According to WolframAlpha, $i^i=e^{-\pi/2}$ but I don't know how I can prove it.
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### How do I get the square root of a complex number?

If I'm given a complex number (say $9 + 4i$), how do I calculate its square root?
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### What is the value of $1^i$?

What is the value of $1^i$? $\,$
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### Wild automorphisms of the complex numbers

I read about so called "wild" automorphisms of the field of complex numbers (i.e. not the identity nor the complex conjugation). I suppose they must be rather weird and I wonder whether someone could ...
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### What is $\sqrt{i}$?

If $i=\sqrt{-1}$, is $\large\sqrt{i}$ imaginary? Is it used or considered often in mathematics? How is it notated?
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### Proving $\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$

I am being asked to prove that $$\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$$ I have some progress made, but I am stuck and could use some help. What I did: ...
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### “Where” exactly are complex numbers used “in the real world”?

I've always enjoyed solving problems in the complex numbers during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be ...
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### How to raise a complex number to the power of another complex number?

How do I calculate the outcome of taking one complex number to the power of another, ie $\displaystyle {(a + bi)}^{(c + di)}$?
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### Prove if $|z| < 1$ and $|w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w} \over {1-zw^*}}| < 1$

Prove if $|z| < 1$ and $|w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w} \over {1-zw^*}}| < 1$Given that $|1-zw^*|^2 - |z-w|^2 = (1-|z|^2)(1-|w|^2)$I think the first part can be proven by ...
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### Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1&0\end{smallmatrix}\right)$ to $i$

I was reviewing some matrices and found this interesting if $r = \begin{pmatrix} 0&1\\ -1&0 \end{pmatrix}$ then $rr=-I$, also $$\exp{(\theta r)} = \cos\theta I + \sin\theta r$$ No wonder, the ...
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### What's the name for the property of a function $f$ that means $f(f(x))=x$?

I can think of several examples of functions such that twice application of the function is equivalent to no application of it. Additive inverse Multiplicative inverse Fourier transform Complex ...
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### Simple Complex Number Problem: $1 = -1$ [duplicate]

Possible Duplicate: -1 is not 1, so where is the mistake? I'm trying to understand the exact point of failure in the following reasoning: \begin{equation*} 1 = \sqrt{1} = \sqrt{(-1)(-1)} = \...
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### What's the difference between $\mathbb{R}^2$ and the complex plane?

I haven't taken any complex analysis course yet, but now I have this question that relates to it. Let's have a look at a very simple example. Suppose $x,y$ and $z$ are the Cartesian coordinates and ...
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### Is “$a + 0i$” in every way equal to just “$a$”?

I'm having a little argument with my friend. He says that "$a + 0i$" is, in every way, absolutely equal to "$a$" (e.g.: $2 + 0i = 2$). I say this is practically the case, so in every calculation you ...
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### Refining my knowledge of the imaginary number

So I am about halfway through complex analysis (using Churchill amd Brown's book) right now. I began thinking some more about the nature and behavior of $i$ and ran into some confusion. I have seen ...